Overspill

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This article is about mathematics. For housing estates, see overspill estate.

In non-standard analysis, a branch of mathematics, overspill is a widely used proof technique. It is based on the fact that the set of standard natural numbers N is not an internal subset of the internal set *N.

Indeed, by applying the induction principle for the standard integers N and the transfer principle we get the principle of internal induction:

For any internal subset A of *N, if

  1. 1 is an element of A and
  2. for every element n of A, n+1 also belongs to A

then

A= *N

If N were an internal set, then instantiating the internal induction principle with N, it would follow N=*N which we know not to be the case.

The overspill principle has a number of extremely useful consequences:

  • The set of standard hyperreals is not internal.
  • The set of bounded hyperreals is not internal.
  • The set of infinitesimal hyperreals is not internal.

In particular:

  • If an internal set contains all infinitesimal non-negative hyperreals, it contains a positive non-infinitesimal (or appreciable) hyperreal.
  • If an internal set contains N it contains an unbounded element of *N.

[edit] Example

We can use these facts to prove equivalence of the following two conditions for an internal hyperreal-valued function f defined on *R.

\forall \epsilon >\!\!\!> 0, \exists \delta >\!\!\!> 0, |h| \leq \delta \implies |f(x+h) - f(x)| \leq \epsilon

and

\forall h \cong 0, \ |f(x+h) - f(x)| \cong 0

The proof that the second fact implies the first uses overspill, since given a non-infinitesimal positive ε

\forall \mbox{ positive } \delta \cong 0, \ (|h| \leq \delta \implies |f(x+h) - f(x)| < \epsilon)

By overspill a positive appreciable δ with the requisite properties exists.

These equivalent conditions express the property known in non-standard analysis as S-continuity of f at x. S-continuity is referred to as an external property, since its extension (e.g. the set of pairs (f, x) such that f is S-continuous at x) is not an internal set.